I really like Lucas' correction and I feel like I understand what he's doing.
I'm fairly sure he is using those cross lines to find intersecting points that he can use as a reference to figure out the symmetry.
If you take a look at the bottom of my little sketch, you'll notice that the area of 1 and 2 are the same, but once it's in perspective it appears as if the area of 2 is greater than area of 1.
In a 2D space that would be true, but in a 3D space like we're working in, the area of 1 and 2 will still be the same.
So in order to keep the area of those two sides the same, Lucas used all of those cross lines and horizonal lines to find intersecting points that he can reference off of to figure out how to correctly draw the arch of the chest in 3D space.
I hope this is right haha. I'm very curious how Lucas approaches his perspective so I'll stick again and await his response as well! :o